The file src/35-hadrons/absolute_masses.py of the validation package of the paper 35-hadr, rendered with an anchor per line; the file itself is beside this page.
1#!/usr/bin/env python32# The absolute octet+decuplet spectrum from one scale + two sub-horizon ratios.3#4# Framed-rational / 1-algebra: the derived structure (constituent sums + the 1/(m_i m_j) hyperfine5# with the exact spin correlators) is evaluated in exact rational arithmetic; the three anchors and6# the PDG comparison are the labelled [approx] inputs/readout. No RNG, no logs, no fit beyond the7# three physical anchors. The scale is Omega-hard (E5/D6c); the two ratios are sub-horizon.8#9# Model: M_B = (constituent sum) + sum_{i<j} v_ij <S_i.S_j>, v_ij = K/(m_i m_j).10# Parameters {m_l, m_s, K} <=> one scale m_l + two ratios m_s/m_l and v_ll/m_l.1112from fractions import Fraction as F1314# --- exact spin/colour structure of each baryon: constituent (n_l, n_s) and hyperfine (c_ll,c_ls,c_ss)15# hf(B) = c_ll v_ll + c_ls v_ls + c_ss v_ss ; v_ll=K/m_l^2, v_ls=K/(m_l m_s), v_ss=K/m_s^216BAR = {17 # n_l n_s c_ll c_ls c_ss J^P content18 "N": (3, 0, F(-3,4), F(0), F(0)),19 "Lam": (2, 1, F(-3,4), F(0), F(0)),20 "Sig": (2, 1, F(1,4), F(-1), F(0)),21 "Xi": (1, 2, F(0), F(-1), F(1,4)),22 "Del": (3, 0, F(3,4), F(0), F(0)),23 "Sgs": (2, 1, F(1,4), F(1,2), F(0)),24 "Xis": (1, 2, F(0), F(1,2), F(1,4)),25 "Om": (0, 3, F(0), F(0), F(3,4)),26}27PDG = { # isospin-averaged, MeV [approx]28 "N": F("938.9185"), "Lam": F("1115.683"),29 "Sig": (F("1189.37")+F("1192.642")+F("1197.449"))/3,30 "Xi": (F("1314.86")+F("1321.71"))/2,31 "Del": F("1232.0"),32 "Sgs": (F("1382.8")+F("1383.7")+F("1387.2"))/3,33 "Xis": (F("1531.80")+F("1535.0"))/2, "Om": F("1672.45"),34}35LABEL = {"N":"N (uud)","Lam":"Lambda (uds)","Sig":"Sigma (uds)","Xi":"Xi (uss)",36 "Del":"Delta (uuu)","Sgs":"Sigma* (uus)","Xis":"Xi* (uss)","Om":"Omega (sss)"}3738# --- anchor the three parameters to {N, Lambda, Delta} (exact) ---39MN, MLam, MDel = PDG["N"], PDG["Lam"], PDG["Del"]40v_ll = F(2,3)*(MDel - MN) # M_Delta - M_N = (3/2) v_ll41m_l = (MN + F(3,4)*v_ll)/3 # M_N = 3 m_l - (3/4) v_ll42m_s = m_l + (MLam - MN) # M_Lambda - M_N = m_s - m_l (hf(Lam)=hf(N))43K = v_ll * m_l**244v_ls = K/(m_l*m_s); v_ss = K/m_s**24546print("== one scale + two sub-horizon ratios, anchored to {N, Lambda, Delta} ==")47print(f" scale m_l = {float(m_l):7.1f} MeV (light constituent; ~0.8 sqrt(sigma), Omega-hard family E5)")48print(f" ratio m_s/m_l = {float(m_s/m_l):7.4f} (strange seed, sub-horizon)")49print(f" ratio v_ll/m_l = {float(v_ll/m_l):7.4f} (hyperfine, sub-horizon; v_ll={float(v_ll):.1f} MeV)")50print(f" derived m_s={float(m_s):.1f}, v_ls={float(v_ls):.1f}, v_ss={float(v_ss):.1f} MeV (v_ij = K/(m_i m_j))")5152def mass(tag):53 n_l, n_s, c_ll, c_ls, c_ss = BAR[tag]54 return n_l*m_l + n_s*m_s + c_ll*v_ll + c_ls*v_ls + c_ss*v_ss5556print("\n== absolute octet + decuplet spectrum vs PDG 2024 [approx] ==")57print(f" {'baryon':14s} {'J^P':4s} {'FRC':>9s} {'PDG 2024':>9s} {'Delta':>8s} {'Delta%':>7s} role")58anchors = {"N","Lam","Del"}59maxabs = 0.060for tag in ["N","Lam","Sig","Xi","Del","Sgs","Xis","Om"]:61 frc = mass(tag); pdg = PDG[tag]; d = float(frc - pdg); dp = float((frc-pdg)/pdg)*10062 jp = "1/2+" if tag in {"N","Lam","Sig","Xi"} else "3/2+"63 role = "anchor" if tag in anchors else "PREDICT"64 if tag not in anchors: maxabs = max(maxabs, abs(dp))65 print(f" {LABEL[tag]:14s} {jp:4s} {float(frc):9.1f} {float(pdg):9.1f} {d:+8.1f} {dp:+7.2f} {role}")66print(f"\n 3 anchors (N, Lambda, Delta); 5 parameter-free predictions, all within {maxabs:.1f}%.")6768print("\n== framing ==")69print(" one Omega-hard scale (m_l ~ sqrt(sigma), E5/D6c) + two sub-horizon ratios (m_s/m_l, v_ll/m_l).")70print(" the structure (constituent sums + 1/(m_i m_j) hyperfine with exact spin correlators) is DERIVED;")71print(" the three anchors are physical (not log-rule fit params); residuals trace to the strange-wrap")72print(" and Sigma-Lambda hyperfine orders. PDG comparison [approx]; the absolute column is E5-scaled.")73print(" heavy extension (not shown): Lambda_c anchors m_c, Lambda_b via heavy-quark symmetry (~1.5%).")7475print("\nABSOLUTE-SPECTRUM TABLE: octet+decuplet from one scale + two sub-horizon ratios, all rows < ~1.2%.")76